Theory of promonads

A promonad 'in' category $\bf D$ is the categorical correspondent of an algebra over a ring $R$.

I prefer to look at profunctors $\ F:{\bf A}^{op}\times{\bf B}\to\bf{Set}\ $ as their collages: considering the elements of each $F(A,B)$ as 'outer arrows' (heteromorphisms) from $A$ to $B$, thus giving a bigger category which contains (disjointly) $\bf A$ and $\bf B$ and these heteromorphisms.

The heteromorphisms of the composition $F\odot G$ are just the consecutive pairs of heteromorphisms (quotienting out by $\langle f\beta,g\rangle\,\sim\,\langle f,\beta g\rangle$ for $\beta$ in the middle category, as a kind of tensor product).

Then, an endoprofunctor $F:{\bf D}\not\to {\bf D}$ contains the same objects as $\bf D$ (each one twice), and possibly heteromorphisms among them. A monoid structure adds an associative composition operation $F\odot F\to F$, and an 'insertion' $\hom_{\bf D}\to F$, representing all original $\bf D$-arrows as actual heteromorphisms in $F$ (note that this is not required to be injective). This yields to a category $\bf F$ on objects of $\bf D$ with the heteromorphisms of $F$ as arrows, equipped with the 'insertion' functor ${\bf D}\to{\bf F}$.

So, in this view, a promonad over $\bf D$ is just another category $\bf F$ on the same object class equipped with a functor $U:\bf D\to\bf F$, which is identical on the objects:

If such is given, form the profunctor $F$ so that $F(A,B):={\bf F}(A,B)$, for morphisms $\gamma:A'\to A,\ \ \delta:B\to B'$, define $$\ F(\gamma,\delta):= f\mapsto U\delta\circ f\circ U\gamma\,,$$ the monoid structure is coming from the composition in $\bf F$, plus $U$ as the unit.

I prefer to think of profunctors $\mathbb{A} \to \mathbb{B}$ (which for me means a functor $\mathbb{A} \to [\mathbb{B}^{\textrm{op}}, \textbf{Set}]$) as secretly being colimit-preserving functors $[\mathbb{A}^\textrm{op}, \textbf{Set}] \to [\mathbb{B}^\textrm{op}, \textbf{Set}]$. Indeed, the presheaf topos $[\mathbb{A}^\textrm{op}, \textbf{Set}]$ is characterised by the following universal property:

• For every cocomplete category $\mathcal{E}$, the functor $\textbf{Cocont}([\mathbb{A}^\textrm{op}, \textbf{Set}], \mathcal{E}) \to [\mathbb{A}, \mathcal{E}]$ sending a colimit-preserving functor $F : [\mathbb{A}^\textrm{op}, \textbf{Set}] \to \mathcal{E}$ to the composite $F Y : \mathbb{A} \to \mathcal{E}$, where $Y : \mathbb{A} \to [\mathbb{A}^\textrm{op}, \textbf{Set}]$ is the Yoneda embedding, is fully faithful and surjective on objects.

In particular, $[\mathbb{B}^\textrm{op}, \textbf{Set}]$ is a cocomplete category, so the category of functors $\mathbb{A} \to [\mathbb{B}^\textrm{op}, \textbf{Set}]$ is pseudonaturally equivalent to the category of colimit-preserving functors $[\mathbb{A}^\textrm{op}, \textbf{Set}] \to [\mathbb{B}^\textrm{op}, \textbf{Set}]$, and one can check that this makes the bicategory of small categories and profunctors 3-equivalent to the 2-category of presheaf toposes and colimit-preserving functors. (One may think of the bicategory of profunctors as being a bicategorical analogue of the Kleisli category for the pseudomonad that sends a small category to its free cocompletion... but such a pseudomonad cannot be defined on the 2-category of small categories, so this is not literally true.)

Thus, a monad on $\mathbb{A}$ in the bicategory of profunctors is the same thing as a monad on $[\mathbb{A}^\textrm{op}, \textbf{Set}]$ whose underlying endofunctor preserves colimits. It is certainly true that every category $\mathbb{B}$ that admits an identity-on-objects functor $\mathbb{A} \to \mathbb{B}$ induces such a monad: indeed, by precomposition, we get a conservative functor $[\mathbb{B}^\textrm{op}, \textbf{Set}] \to [\mathbb{A}^\textrm{op}, \textbf{Set}]$ with a left and right adjoint, and thus by Beck's monadicity theorem, $[\mathbb{B}^\textrm{op}, \textbf{Set}]$ is monadic over $[\mathbb{A}^\textrm{op}, \textbf{Set}]$.

On the other hand, suppose we have a monad $\mathsf{T}$ on $[\mathbb{A}^\textrm{op}, \textbf{Set}]$ whose underlying endofunctor preserves colimits. Let $\mathcal{B}$ be the category of $\mathsf{T}$-algebras. By standard nonsense, the forgetful functor $U : \mathcal{B} \to [\mathbb{A}^\textrm{op}, \textbf{Set}]$ is conservative, has a left adjoint $F : [\mathbb{A}^\textrm{op}, \textbf{Set}]$, and creates both limits and colimits. Using Weber's nerve theorem, it follows that the full subcategory $\mathbb{B}$ of $\mathcal{B}$ spanned by the $\mathsf{T}$-algebras of the form $F Y A$ for $A$ an object in $\mathbb{A}$ has the property that the restricted Yoneda embedding $\mathcal{B} \to [\mathbb{B}^\textrm{op}, \textbf{Set}]$ is fully faithful and preserves all colimits. But the universal property of $[\mathbb{B}^\textrm{op}, \textbf{Set}]$ implies that this is an equivalence of categories, so in fact every monad whose underlying endofunctor preserves colimits must be of this form.

Before we start some general facts about cocomplete monoidal closed category:

• in any category of this kind $\mathcal V$ we can build up the bicategory of monoid, bimodules and bimodules morphisms;
• fixed any monoid $M$ in this bicategory we can consider the monoidal category of $M,M$-bimodules;
• in such monoidal category a monoid is just a monoid $N$ in $\mathcal V$ with a monoid morphism $M \to N$.

This last fact is the generalization of the well know fact about $R$-algebras, i.e. that every $R$-algebra can be viewed as an $R,R$-bimodule with a $R$-linear associative and unital multiplication or as a ring with a ring homomorphism from $R$ (we remind that rings are monoid in the cocomplete closed monoidal category $\mathbf {Ab})$.

Now let's do some work!

John Baez told that we can recover the bicategory of sets, spans between set and span morphisms as the bicategory of monoids, bimodules and bimodules morphisms on $\mathbf {Set}^\text{op}$.

A monad in such bicategory is nothing more than a category (as you stated in the question) and a bimodule is just a profunctor.

So the bicategory of categories, profunctors and natural transformation between those is nothing but the bicategory of monoids, bimodules and bimodules morphisms in $\mathbf {Set}^\text{op}$.

Ok now back to your question: what is a monad in such bicategory? Well for start a monad is a monoid in one of the monoidal category of $C,C$-bimodules (a.k.a. profunctors).

A monad should just be a $C,C$-bimodule for some category/monoid $C$, with a $C$-linear monoidal structure.

From what we have said above this is data are equivalent to a monoid $N$ in the monoidal category of spans in $\mathcal{O}(C)$ (so $N$ is just a category) with a monoid morphism (i.e. a functor) from the category $C$ to $N$.

Hope this helps.